OPTIMAL STOCKPILES UNDER STOCHASTIC UNCERTAINTY

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Abstract

We study stockpiling problems under uncertain economic and physical factors, and investigate the valuation and optimisation of storage systems where the availability and spot price of the underlying are both subject to stochasticity. Following a Real Options valuation approach, we first study financial derivatives linked to Asian options. A comprehensive set of boundary conditions is compiled, and an alternative (and novel) similarity reduction for fixed-strike Asian options is derived. Hybrid semi-Lagrangian methods for numerically solving the related partial differential equations (PDEs) are implemented, and we assess the accuracy of the valuations thus obtained with respect to results from classical finite-difference valuation methods and with respect to high precision calculations for valuing Asian options with spectral expansion theory techniques. Next we derive a PDE model for valuing the storage of electricity from a wind farm, with an attached back-up battery, that operates by trading electricity in a volatile market in order to meet a contracted fixed rate of energy generation; this system comprises two diffusive-type (stochastic) variables, namely the energy production and the electricity spot price, and two time-like (deterministic) variables, specifically the battery state and time itself. An efficient and novel semi-Lagrangian alternating-direction implicit (SLADI) methodology for numerically solving advection-diffusion problems is developed: here a semi-Lagrangian approach for hyperbolic problems of advection is combined with an alternating-direction implicit method for parabolic problems involving diffusion. Efficiency is obtained by solving (just) tridiagonal systems of equations at every time step. The results are compared to more standard semi-Lagrangian Crank-Nicolson (SLCN) and semi-Lagrangian fully implicit (SLFI) methods. Once he have established our PDE model for a storage-upgraded wind farm, a system that depends heavily on the highly stochastic nature of wind and the volatile market where electricity is sold, we derive a Hamilton-Jacobi-Bellman (HJB) equation for optimally controlling charging and discharging rates of the battery in time, and we assess a series of operation regimes. The solution of the related PDE models is approached numerically using our SLADI methodology to efficiently treat this mixed advection and diffusion problem in four dimensions. Extensive numerical experimentation confirms our SLADI methodology to be robust and yields highly accurate solutions and efficient computations, we also explore effects from correlation between stochastic electricity generation and random prices of electricity as well as effects from a seasonal electricity spot price. Ultimately, the objective of approximating optimal storage policies for a system under uncertain economic and physical factors is accomplished. Finally we examine the steady-state solution of a stochastic storage problem under uncertain electricity market prices and fixed demand. We use a HJB formulation for optimally controlling charging and discharging rates of the storage device with respect to the electricity spot price. A projected successive over-relaxation coupled with the semi-Lagrangian method is implemented, and we explore the use of boundary-fitted coordinates techniques.

Keyword(s)

Alternating-direction; Asian option; Optimisation; Semi-Lagrangian; Similarity reduction; Stochastic storage

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